The methods to deal with missing data, implicitly assume a missing data mechanism.

MCAR: the most strict assumption. In practice it is also easiest to deal with MCAR data.

1. Analyze the observed sample only (this will result in unbiased estimates).
2. Use an imputation method to boost the power the the amount of missing data is too large.

MAR: less strict assumption. Most advanced missing data methods assume this mechanism (e.g. multiple imputation, FIML).

• Include variables in study that may explain the missing data, a MAR assumption may become more plausible (as compared to MNAR).
• These auxiliary variables may also help in dealing with the missing data.
• Auxiliary variables: variables related to the probability of missing data or to the variable with missing data. Can be used as predictors in an imputation model or as covariates in a FIML model to improve estimations.

MNAR: least strict assumption.

• MNAR data are also referred to as non-ignorable, because these cannot be ignored without causing bias in results. MNAR data are more challenging to deal with.

Complete-case analysis (CCA): only the cases with observed data for all variables involved are used in the analysis.

• The most common and easy way to deal with missing data.
• The default methods for many data analysis procedure.
• Assumes MCAR

Parameter estimates:

• Mean is unbiased for MCAR
• Regression weight/correlation unbiased for MCAR
• Standard errors are overestimated

## # A tibble: 2 x 5
##   term        estimate std.error statistic p.value
##   <chr>          <dbl>     <dbl>     <dbl>   <dbl>
## 1 (Intercept)    0.347    0.205       1.69  0.0935
## 2 X2             0.225    0.0922      2.44  0.0167

## # A tibble: 2 x 5
##   term        estimate std.error statistic p.value
##   <chr>          <dbl>     <dbl>     <dbl>   <dbl>
## 1 (Intercept)    0.418     0.325      1.29   0.204
## 2 X2             0.245     0.170      1.44   0.157

## # A tibble: 2 x 5
##   term        estimate std.error statistic p.value
##   <chr>          <dbl>     <dbl>     <dbl>   <dbl>
## 1 (Intercept)    0.347    0.205       1.69  0.0935
## 2 X2             0.225    0.0922      2.44  0.0167

## # A tibble: 2 x 5
##   term        estimate std.error statistic p.value
##   <chr>          <dbl>     <dbl>     <dbl>   <dbl>
## 1 (Intercept)  -0.0302     0.335   -0.0900   0.929
## 2 X2            0.0942     0.150    0.627    0.534

• Most important: assumes that the imputed value is an actual observed value.
• Some assume MCAR (i.e. regression imputation, random value imputation)

Other examples:

• All missing values are average (i.e. mean imputation)
• People with missing data remain stable (i.e. LOCF)

• Imputing the average value for all missing entries
• Very easy method

Parameter estimates:

• Mean is unbiased when MCAR
• Regression weight/correlation never unbiased
• Standard errors are underestimated

Conditional mean imputation, estimating the imputed value as a predicted value from a regression model: $Y_{imp} = _0 + _1 * X

• Mean is unbiased when MAR
• Regression weight is unbiased when MAR
• Correlation is never unbiased
• Standard errors are underestimated

Imputation from regression equation: \(Performance = \beta_0 + \beta_1 * weight + \beta_2 * gender\)

Stochastic regressions: = regression imputation, with additional sampling error added to the predicted value:

• \(Y_{imp} = \beta_0 + \beta_1 * X + \epsilon\)

Sampling error is normally distributed.

• Mean is unbiased when MAR
• Regression weight is unbiased when MAR
• Correlation is unbiased when MAR
• Standard errors are underestimated

Imputation uncertainty is not taken into account

Imputation from regression equation: \(Performance = \beta_0 + \beta_1 * weight + \beta_2 * gender + \epsilon\)

Ad hoc method for longitudinal data: use the previous observed value to impute the missing values.

Assumes that people that drop out of the study remain stable.

• Mean, regression coefficient and correlation is never unbiased.
• Standard errors are underestimated

The trajectories over time, with LOCF imputations in red.

Below only the cases with missing observations, separated by group.

• Imputing the missing data entries with multiple “plausible” values.
• Takes imputation uncertainty into account
• A method to improve the main analysis results, (so not to complete or fill in data)
  • Unbiased estimates
  • Improved precision and thus power
• R package: mice

• Imputation phase
• Analysis phase
• Pooling phase

1. incomplete data
2. generate multiple copies of the same dataset, but each time differenty imputed values
3. analyze each imputed dataset
4. pool results for analyses to final study result

• Imputing the missing data entries with multiple “plausible” values
• Specify the imputation model
  • method to estimate imputed values
  • predictors used in the model

• Fully Conditional Specification (FCS): imputes variable-by-variable.
• For each variable the imputation model can be different:
  • method
  • predictors
• Joint modelling: imputes from a multivariate distribution

• Impute missing value by using the predicted value estimated from observed data.
• Add residual error to simulate sampling error
• Depending on the variable type that needs to be imputed, the imputation method is defined.

Later more on methods

Imputation model

• For each variable, the most relevant predictors can be indicated
• Use variables that are in the analysis model also in the imputation model (compatibility)
• Add auxiliary variables
• Advice to use no more than 25 predictors (also depends on sample size)

Auxiliary variables: variables related to the probability of missing data or to the variable with missing data.

• One iteration, consists of one cycle through all variables that need to be imputed.
  • The algorithm starts with a random draw from the observed data.
  • Then imputes the incomplete data in a variable-by-variable fashion.
• Typically 5 to 10 iterations before an imputed dataset is generated.
• Iterations are repeated to ensure convergence of the algorithm

• To check the convergence of the algorithm

• Use additional iterations when in doubt if convergence is reached

• Convergence may need more iterations

• Looks better for gen and tv, but there seems to be a problem for wgt.

• Changes in imputation model may be needed when:
• Two variables are highly correlated (multi-collinearity)
• A variable with so many missings that the imputations are unstable

• Theoretically, higher \(m\) is better.
• Usually, use \(m = 5\) for model building and \(m ~ % missing data\) for final analysis.
• In lower \(m\) the between-imputation-variance is enlarged by \(1/m\) before computing the total variance.

• Each imputed dataset is analyzed, with the substantive analysis model

• This results in \(m\) sets of results

• Workflow:

  • Imputation
  • Analysis
  • Pooling

• Pool the analysis results to obtain final parameter estimates.
• Usual normally distributed parameters: Rubin’s Rules

## Class: mipo    m = 5 
##          term m   estimate         ubar            b            t dfcom
## 1 (Intercept) 5 21.5600906 3.257524e+01 1.803273e+00 3.473916e+01   151
## 2     Solar.R 5  0.1124339 7.638285e-04 4.843572e-05 8.219513e-04   151
##         df        riv     lambda        fmi
## 1 123.0708 0.06642860 0.06229072 0.07716663
## 2 118.0594 0.07609413 0.07071327 0.08606584

• riv: Relative increase in variance due to nonresponse
• df Residual degrees of freedom for hypothesis testing
• lambda Proportion of total variance due to missingness
• fmi Fraction of missing information

• Pooling of point estimates that are normally distributed over the imputed datasets.

• Means, standard deviations, regression estimates, linear predictors, proportions.

• For pooling point estimates, use mean:

  \(\hat\theta = \sum^m_{i=1}{\theta_i}\)

• Pooling of variance or standard error around the estimate, combine the within and between imputation variance.

• Between variance:

  \(\sigma^2_{between} = \frac{\sum^m_{i=1}(\beta_i - \overline\beta)^2}{m-1}\)

• Within variance:

  \(\sigma^2_{within} = \frac{\sum^m_{i=1}\sigma^2_i}{m}\)

• Total variance:

  \(\sigma^2_{total} = \sigma^2_{within} + \sigma^2_{between} + \frac{\sigma^2_{between}}{m}\)

• Use a transformation to approximate a normal distribution
• Examples:
  • Correlation: Fisher \(z\)
  • Odds Ratio, Relative Risk, Hazard ratio: log transformation
  • Explained variance: Fisher \(z\) for \(\sqrt{R^2}\)

• Imputation phase
• Analysis phase
• Pooling phase

In the imputation phase, imputed values are estimated using an imputation method.

• Regression imputation method = norm.predict
• Stochastic regression imputation (regression + residual error) method = norm.nob
• Stochastic regression with parameter uncertainty method = norm
• Predictive mean matching method = pmm

• Imputed value: \(Y_{imp} = \hat{\beta}_0 + X_{mis}\hat{\beta}_1 + \epsilon\)

• Parameters \(\hat{\beta}_0\) and \(\hat{\beta}_1\) are estimated from the observed data.

• Imputed value: \(Y_{imp} = \hat{\beta}_0 + X_{mis}\hat{\beta}_1 + \epsilon\)

• Parameters \(\hat{\beta}_0\) and \(\hat{\beta}_1\) are estimated from the observed data.

• \(\epsilon\) is normally distributed residual error

• Imputed value: \(Y_{imp} = \dot{\beta}_0 + X_{mis}\dot{\beta}_1 + \epsilon\)

• Parameters \(\dot{\beta}_0\) and \(\dot{\beta}_1\) are drawn from their posterior distribution.

• \(\epsilon\) is normally distributed residual error

• Estimate the predicted value using Bayesian regression imputation
  • \(Y_{imp} = \dot{\beta}_0 + X_{mis}\dot{\beta}_1 + \epsilon\)
  • Parameters \(\dot{\beta}_0\) and \(\dot{\beta}_1\) are drawn from their posterior distribution.
  • \(\epsilon\) is normally distributed residual error
• Select \(k\) nearest neighbors to this predicted value from the observed data
• Randomly draw one donor to use as imputed value

• Bayesian logistic regression method = logreg
• Bayesian Polytomous regression method = polyreg
• Classification and regression trees method = cart

• Imputed value: \(Log\frac{P(Y_{miss})}{1-P(Y_{mis})} = \dot{\beta}_0 + X_{mis}\dot{\beta}_1 + \epsilon\)

• Parameters \(\dot{\beta}_0\) and \(\dot{\beta}_1\) are drawn from their posterior distribution.

• \(\epsilon\) is normally distributed residual error

• Fit a multinomial regression model.

• Parameters are drawn from their posterior distribution (Bayesian).

• Compute the predicted category.

• Add normally distributed residual error to account for sampling variance.

• Draw a random bootstrap sample as training set.
• Fit the tree model using the observed data.
• Find the predicted terminal node for the missing value
• Use the observed cases at the predicted terminal node as donors
• Randomly draw the imputed value from the observed donor cases